Sobolev and Hardy type inequalities play an important role in many areas of mathematics and mathematical physics. They have become standard tools in existence and regularity theories for solutions to partial differential equations, in calculus of variations, in geometric measure theory and in stability of matter. In analysis a number of inequalities like the Hardy–Littlewood– Sobolev inequality...

Let $\Omega$ be an open set in a metric measure space $X$. Our main result gives equivalence between the validity of weighted Hardy–Sobolev inequality and quasiadditivity capacity with respect to Whitney covers $\Omega$. Important ingredients proof include use discrete convolution as test function Maz'ya type characterization inequalities.

Inequalities of the form ∑∞ k=0 |f̂(mk)| k+1 ≤ C ‖f‖1 for all f ∈ H1, where {mk} are special subsequences of natural numbers, are investigated in the vector-valued setting. It is proved that Hardy’s inequality and the generalized Hardy inequality are equivalent for vector valued Hardy spaces defined in terms of atoms and that they actually characterize B-convexity. It is also shown that for 1 < ...

(1.1) and (1.2) is the well known Hardy-Littlewood-Polya’s inequality. In connection with applications in analysis, their generalizations and variants have received considerable interest recent years. Firstly, by means of introducing a parameter, two forms of extended Hardy-Littlewood-Polya’s inequality are obtained by Hu in [2] as follows. (1) Let λ > 0, p > 1, 1 p+ 1 q=1, f(x), g(y) ≥ 0, F (x...

In 1986 A. Ancona showed, using the Koebe one-quarter Theorem, that for a simply-connected planar domain the constant in the Hardy inequality with the distance to the boundary is greater than or equal to 1/16. In this paper we consider classes of domains for which there is a stronger version of the Koebe Theorem. This implies better estimates for the constant appearing in the Hardy inequality. ...

In this paper, we establish a general weighted Hardy type inequality for the $ p- $Laplace operator with Robin boundary condition. We provide various concrete examples to illustrate our results different weights. Furthermore, present some Heisenberg-Pauli-Weyl inequalities terms on balls centred at origin radius R in \mathbb{R} ^n $.

In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of [Formula: see text], all them involving a mean curvature term and having universal constants independent the hypersurface. We first consider celebrated Sobolev inequality Michael–Simon Allard, in our codimension one framework. Using their ideas, but simplifying presentations, give quick easy-to-read proof inequality. Next, e...

In this paper, we consider radial symmetry property of positive solutions of an integral equation arising from some higher order semi-linear elliptic equations on the whole space Rn. We do not use the usual way to get symmetric result by using moving plane method. The nice thing in our argument is that we only need a Hardy-Littlewood-Sobolev type inequality. Our main result is Theorem 1 below.

In this paper, following a new interpolation approach in fixed point theory, we introduce the concepts of interpolative Hardy-Rogers-type fuzzy contraction and Reich-Rus-Ciric type framework metric spaces, analyze existence points for such contractions equipped with some suitable hypotheses. A few consequences single-valued mappings which include conclusion main result Karapinar et al. [On Hard...